Why is there a white circle when I try to move vertices?
The white circle is due to "Proportional Editing" being turned on. It allows artist to move one element - a vertex - and a falloff is applied to the adjacent elements - neighbouring vertices. You will only need to turn "Proportional editing" off to stop the other neighbouring vertices from getting transformations you don't need.

It can be toggle on or off by pressing O. Alternatively, the button is here:



The button appears in the UV editor menu as well. They're linked, turning on the button on in the 3D view will also turn it on inside the UV editor.